Let X and Y be Bernoulli random variables. Let Z = X+ Y.

a. Show that if X and Y cannot both be equal to 1, then Z is a Bernoulli random variable.

b. Show that if X and Y cannot both be equal to 1, then pz=px+py

c. Show that if X and Y can both be equal to 1, then Z is not a Bernoulli random variable.

Answer:

Step 1 of 3:

(a)

In this question, we are asked to prove that if and cannot both be equal to 1, then is a Bernoulli random variable.

Let and be Bernoulli random variables. Let = .

Case 1:

If and ,

Case 2:

If and ,

So in both the cases .

The random variable is said to have the Bernoulli distribution when random event results in the success, then . Otherwise .

So from the definition we can say that is also a Bernoulli random variable.

Since value of .

Hence proved.

Step 2 of 3:

(b)

In this question, we are asked to prove that if and cannot both be equal to 1, then

From the addition rule of probability, we have

= ……….(1)

Since and cannot both be equal to 1.

Hence ,

we can rewrite equation (1) as,

=

, and

Substitute above values, we have

Hence proved.